One millisievert is exactly 0.001 sievert. The millisievert is the unit radiation protection actually works in, because a whole sievert is a dose nobody should ever receive — the prefix is the difference between a routine year and an emergency.
About the units
The sievert is the SI unit of equivalent and effective dose. It is a large unit relative to anything encountered in normal life: natural background radiation gives a person about 2 to 3 mSv per year in most of the world, and up to 7 or 8 in areas with radon-rich geology. Acute radiation sickness begins around 1 Sv delivered over a short period, which is roughly four hundred years of background compressed into an afternoon. Because of that range, doses are recorded in millisieverts and microsieverts, and the unprefixed sievert appears mainly in accident analysis and radiotherapy side-effect estimates.
The exact factor
The factor is exactly 1,000. Reference values worth having: natural background 2 to 3 mSv per year; a transatlantic flight 0.05 mSv; a chest X-ray 0.02 mSv; a chest CT 6 to 7 mSv; the EU occupational limit 20 mSv in any single year; the public limit 1 mSv per year above background. Radon in poorly ventilated buildings is the largest variable contributor in Sweden and much of northern Europe, and can add several millisieverts annually on its own.
Where you meet this conversion
Occupational dose monitoring, medical exposure records and public communication about radiation risk. Personal dosimeters read in millisieverts, and cumulative annual dose is the regulated quantity. Nuclear medicine procedures are quoted in effective dose in millisieverts so they can be compared with each other and with background. Aircrew exposure, which comes from cosmic radiation at altitude, is monitored in the same unit and reaches 2 to 5 mSv per year for long-haul crew.